相对量子上同调
Relative quantum cohomology
AI总结：
该研究建立开WDVV偏微分方程系统，定义相对量子上同调代数与联络并证明相关性质，推导壁穿公式与消失定理，通过递推关系确定特定拉格朗日对的开Gromov-Witten不变量，得到计数不变量的显式公式。
摘要：
我们建立了一套名为开WDVV的偏微分方程（PDE）系统，该系统约束了带边界链的拉格朗日子流形$L \subset X$的体形变超势及相关的开Gromov-Witten不变量。同时，我们定义了$X$相对于$L$的量子上同调代数并证明其具有结合性，还定义了相对量子联络并证明其是平坦的。我们推导了一个壁穿公式，该公式允许在开Gromov-Witten不变量中交换点状边界约束与某些内部约束。另一项结果是针对同调非平凡拉格朗日量的开Gromov-Witten不变量的消失定理，当这类不变量带有多于一个点状边界约束时成立。在此情形下，带一个点状边界约束的开Gromov-Witten不变量被证明可恢复某些闭不变量。从开WDVV与壁穿公式出发，我们推导了一组递推关系，它们完全确定了$(X,L) = (\mathbb{C}P^n, \mathbb{R}P^n)$（$n$为奇数）的开Gromov-Witten不变量，这些不变量在作者此前的工作中已有定义。由此，我们得到了利用Fukaya-Oh-Ohta-Ono边界链理论定义的计数不变量的显式公式。
英文摘要：
We establish a system of PDE, called open WDVV, that constrains the bulk-deformed superpotential and associated open Gromov-Witten invariants of a Lagrangian submanifold $L \subset X$ with a bounding chain. Simultaneously, we define the quantum cohomology algebra of $X$ relative to $L$ and prove its associativity. We also define the relative quantum connection and prove it is flat. A wall-crossing formula is derived that allows the interchange of point-like boundary constraints and certain interior constraints in open Gromov-Witten invariants. Another result is a vanishing theorem for open Gromov-Witten invariants of homologically non-trivial Lagrangians with more than one point-like boundary constraint. In this case, the open Gromov-Witten invariants with one point-like boundary constraint are shown to recover certain closed invariants. From open WDVV and the wall-crossing formula, a system of recursive relations is derived that entirely determines the open Gromov-Witten invariants of $(X,L) = (\mathbb{C}P^n, \mathbb{R}P^n)$ with $n$ odd, defined in previous work of the authors. Thus, we obtain explicit formulas for enumerative invariants defined using the Fukaya-Oh-Ohta-Ono theory of bounding chains.
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